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Let D be an integral domain with quotient ÿeld K. A multiplicative subset S of D is a t-splitting set if for each 0 = d ∈ D, dD = (AB)t for some integral ideals A and B of D, where At ∩ sD = sAt for all s ∈ S and Bt ∩ S = ∅. A t-splitting set S of D is a t-lcm (resp., Krull) t-splitting set if sD ∩ dD is t-invertible (resp., sD is a t-product of height-one prime ideals of D) for all nonunits s ∈ S and 0 = d ∈ D. Let S be a t-splitting set of D, T={A1 · · · An | Ai =diDS ∩D for some 0 = di ∈ D}, and D T = {x ∈ K | xC ⊆ D for some C ∈ T}. We show that S is a t-lcm (resp., Krull) t-splitting set if and only if D T is a PVMD (resp., Krull domain), if and only if every ÿnite type integral v-ideal (resp., every integral ideal) of D intersecting S is t-invertible. We also show that D\{0} is a t-splitting set in D[X ] if and only if D is a UMT-domain and that every nonempty multiplicative subset of D[X ] contained in G = {f ∈ D[X ] | (A f )v = D} is a t-lcm t-complemented t-splitting set of D[X ]. Using this, we give several Nagata-like theorems.

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Given a stable semistar operation of finite type $\star$ on an integral domain $D$, we show that it is possible to define in a canonical way a stable semistar operation of finite type $[\star]$ on the polynomial ring $D[X]$, such that $D$ is a $\star$-quasi-Pr\"ufer domain if and only if each upper to zero in $D[X]$ is a quasi-$[\star]$-maximal ideal. This result completes the investigation initiated by Houston-Malik-Mott in the star operation setting. Moreover, we show that $D$ is a Pr\"ufer $\star$-multiplication (resp., a $\star$-Noetherian; a $\star$-Dedekind) domain if and only if $D[X]$ is a Pr\"ufer $[\star]$-multiplication (resp., a $[\star]$-Noetherian; a $[\star]$-Dedekind) domain. As an application of the techniques introduced here, we obtain a new interpretation of the Gabriel-Popescu localizing systems of finite type on an integral domain $D$, in terms of multiplicatively closed sets of the polynomial ring $D[X]$